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Logarithmic Potential Theory with

Applications to Approximation Theory

E.B. Saff ∗

12 October 2010

Abstract

We provide an introduction to logarithmic potential theory in the complex plane that par-ticularly emphasizes its usefulness in the theory of polynomial and rational approximation. Thereader is invited to explore the notions of Fekete points, logarithmic capacity, and Chebyshevconstant through a variety of examples and exercises. Many of the fundamental theorems ofpotential theory, such as Frostman’s theorem, the Riesz Decomposition Theorem, the Principleof Domination, etc., are given along with essential ideas for their proofs. Equilibrium measuresand potentials and their connections with Green functions and conformal mappings are pre-sented. Moreover, we discuss extensions of the classical potential theoretic results to the casewhen an external field is present.

MSC: Primary: 31Cxx, 41Axx

Keywords: Logarithmic potential, Polynomial approximation, Rational approximation, Trans-finite diameter, Capacity, Chebyshev constant, Fekete points, Equilibrium potential, Superhar-monic functions, Subharmonic functions, Green functions, Rates of polynomial and rationalapproximation, Condenser capacity, External fields.

0 Introduction . . . . . . . . . . . . . . . 1661 Transfinite Diameter, Capacity, and Chebyshev Constant . . 1672 Harmonic, Superharmonic and Subharmonic Functions . . . 1783 Equilibrium Potentials, Green Functions and Regularity . . 1834 Applications to Polynomial Approximation of Analytic Functions 1895 Rational Approximation . . . . . . . . . . . 1926 Logarithmic Potentials with External Fields . . . . . . 196References . . . . . . . . . . . . . . . . 199

∗Research was supported, in part, by the U. S. National Science Foundation under grant DMS-0808093.

Surveys in Approximation TheoryVolume 5, 2010. pp. 165–200.c© 2010 Surveys in Approximation Theory.ISSN 1555-578XAll rights of reproduction in any form reserved.

165

E. B. Saff 166

0 Introduction

Logarithmic potential theory is an elegant blend of real and complex analysis that has had aprofound effect on many recent developments in approximation theory. Since logarithmic potentialshave a direct connection with polynomial and rational functions, the tools provided by classicalpotential theory and its extensions to cases when an external field (or weight) is present, haveresolved some long-standing problems concerning orthogonal polynomials, rates of polynomial andrational approximation, convergence behavior of Pade approximants (both classical and multi-point), to name but a few. Here are some problems where potential theory has played a crucialrole:

(i) Rate of Polynomial Approximation: Let f be analytic on a compact set E of the complexplane C, whose complement C \ E is connected. How well can f be uniformly approximatedon E by polynomials (in z) of degree at most n?

(ii) Asymptotic Behavior of Zeros of Polynomials: Let p∗n(x) denote the polynomial of degree atmost n of best uniform approximation to a continuous function on [−1, 1], say f(x) = |x|. Inthe complex plane, p∗n has n zeros (at most).∗ Where are these zeros located as n→ ∞?

(iii) Fast Decreasing Polynomials: Given ϕ ∈ C[−1, 1], does there exist a sequence of “needle-like”polynomials (pn), deg pn ≤ n, such that pn(0) = 1 and |pn(x)| ≤ Ce−cnϕ(x), x ∈ [−1, 1], forsome positive constants c, C?

(iv) Recurrence Coefficients for Orthogonal Polynomials: Let pn denote orthonormal polynomi-als with respect to the weight exp (−|x|α) , α > 0, on R. That is,

∫ ∞

−∞pm(x)pn(x)e−|x|α dx = δmn.

Then the pn satisfy a 3-term recurrence of the form

xpn(x) = an+1pn+1(x) + anpn−1(x), n = 1, 2, . . . ,

where (an) is the sequence of recurrence coefficients. These coefficients go to infinity as nincreases, but exactly what is their asymptotic growth rate?

(v) Generalized Weierstrass Problem: A famous theorem of Weierstrass states that f ∈ C[−1, 1]if and only if there exists a sequence of polynomials (pn), deg pn ≤ n, such that pn → funiformly on [−1, 1]. But how would you characterize those f ∈ C[−1, 1] that are uniformlimits on [−1, 1] of “incomplete” polynomials of the form q2n(x) =

∑2nk=n akx

k, for whichhalf the coefficients are missing? More generally, what functions f are the uniform limitsof weighted polynomials of the form w(x)npn(x), where the power of the weight matches thedegree of the polynomial?

(vi) Optimal Point Arrangements on the Sphere: How well separated are N (≥ 2) points on theunit sphere S2 = x ∈ R

3 : |x| = 1 that maximize the product of their pairwise distances:∏

1≤i<j≤N

|xi − xj | ?

∗By symmetry and uniqueness of the best approximants, p∗2n+1 = p∗

2n for f(x) = |x|.

Logarithmic Potential Theory 167

(vii) Rational Approximation: Determine the rate of best uniform approximation to e−x on [0,+∞)by rational functions of the form pn(x)/qn(x), deg pn ≤ n, deg qn ≤ n.

In this article we provide an introduction to the tools of classical and “weighted” potentialtheory that are the keys to resolving the above questions. The essential reason for the usefulnessof potential theory in obtaining results on polynomials is the fact that for any monic polynomialp(z) =

∏nk=1(z − zk) the function log(1/|p(z)|) can be written as a logarithmic potential:

log1

|p(z)| =

∫

log1

|z − t| dν(t),

where ν is the discrete measure with mass 1 at each of the zeros of p.

1 Transfinite Diameter, Capacity, and Chebyshev Constant

We begin by introducing three “different” quantities associated with a compact (closed and bounded)set in the plane.

A Geometric Problem. Place n points on a compact set E so that they are “as far apart” aspossible in the sense of the geometric mean of the pairwise distances between the points. Since thenumber of different pairs of n points is n(n− 1)/2, we consider the quantity

δn(E) := maxz1,...,zn∈E

∏

1≤i<j≤n

|zi − zj |

2/n(n−1)

. (1.1)

Any system of points Fn :=

z(n)1 , . . . , z

(n)n

for which the maximum is attained is called an n-point

Fekete set for E; the points z(n)i in Fn are called Fekete points.

For example, if n = 2, then F2 =

z(2)1 , z

(2)2

, where∣

∣

∣z(2)1 − z

(2)2

∣

∣

∣ = diam E. Obviously, any

such points lie on the boundary of E. In general, it follows from the maximum modulus principlefor analytic functions that for all n, the Fekete sets lie on the outer boundary ∂∞E, that is, theboundary of the unbounded component of the complement of E.

Exercise. Prove that the determinant of the n × n Vandermonde matrix [zji ], 1 ≤ i ≤ n, 0 ≤

j ≤ n− 1, is given by∏

1≤i<j≤n(zj − zi). Consequently, an n-point Fekete set for E maximizes themodulus of this determinant over all n-point subsets of E.

Exercise. Let E be the closed unit disk (or the unit circle). Prove that the set of nth rootsof unity is an n-point Fekete set for E (and so is any of its rotations) and that δn(E) = n1/(n−1).[Hint: Use Hadamard’s inequality for determinants.]

If E = [−1, 1], then the set Fn turns out to be unique and it coincides with the zeros of(1 − x2)P ′

n−1(x), where Pn−1 is the Legendre polynomial of degree n− 1 (cf. [Sz]).Fekete points are “good points” for polynomial interpolation. We denote by Pn the linear space

of all algebraic polynomials with complex coefficients of degree at most n. Recall that if z1, . . . , zn+1

E. B. Saff 168

are any n + 1 distinct points, then the unique polynomial in Pn that interpolates a function f inthese points is given by

pn(z) =

n+1∑

k=1

f(zk)Lk(z),

where Lk(z) is the fundamental Lagrange polynomial that satisfies Lk(zj) = δjk.

Exercise. Prove that if z1, . . . , zn+1 is an (n+ 1)-point Fekete set for a compact set E, then theassociated fundamental Lagrange polynomials satisfy |Lk(z)| ≤ 1 for all z ∈ E. Furthermore, showthat if Pn ∈ Pn, then

‖Pn‖E ≤ (n+ 1)‖Pn‖Fn+1 ,

where ‖·‖A denotes the sup norm on A.

Exercise. Prove that if Pn(f ; z) denotes the polynomial of degree at most n that interpolatesa continuous function f in an (n+ 1)-point Fekete set for E, then

‖f − Pn(f ; ·)‖E ≤ (n+ 2)‖f − p∗n‖E ,

where p∗n is the best uniform approximation to f on E out of Pn.

On taking the logarithm in (1.1), we see that the max problem in (1.1) is equivalent to theminimization problem

En(E) := minz1,...,zn∈E

∑

1≤i<j≤n

log1

|zi − zj|. (1.2)

The summation in (1.2) can be interpreted as the energy of a system of n like-charged particleslocated at the points zin

i=1, where the repelling force between two particles is proportional to thereciprocal of the distance between them. Thus

En(E) =n(n− 1)

2log

1

δn(E)(1.3)

denotes the minimal logarithmic energy that can be attained by n particles that are constrainedto lie on E. Any set of n points that attains this minimal energy is called an equilibriumconfiguration for E; that is, a Fekete set Fn represents an n-point equilibrium configuration forE.

Essential questions are:

(i) What is the asymptotic behavior of the minimal energy En(E) (or, equivalently, of δn(E)) asn→ ∞?

(ii) How are optimal configurations (Fekete points) distributed on E as n→ ∞?

As a first step we establish

Lemma 1.1. The sequence(

En(E)n(n−1)

)∞

n=2is increasing (i.e., nondecreasing) or, equivalently, the

sequence (δn(E))∞n=2 is decreasing (i.e., nonincreasing).


Proof. With Fn =

z(n)k

n

k=1we have, for each k = 1, . . . , n,

En(E) =∑

i6=k

log1

∣

∣

∣z(n)i − z

(n)k

∣

∣

∣

+∑

1≤i<j≤ni6=kj 6=k

log1

∣

∣

∣z(n)i − z

(n)j

∣

∣

∣

≥∑

i6=k

log1

∣

∣

∣z(n)i − z

(n)k

∣

∣

∣

+ En−1(E).

Now add these n inequalities together and divide by n(n− 1)(n − 2) to get result.

The sequence (δn(E)) therefore has a limit†

τ(E) := limn→∞

δn(E), (1.4)

which is called the transfinite diameter of E. For example, the transfinite diameter of the diskE = z ∈ C : |z| ≤ R is R since δn(E) = Rn1/(n−1) → R as n→ ∞.

Note that 0 ≤ τ(E) ≤ diam E and that E1 ⊂ E2 implies τ(E1) ≤ τ(E2).

Exercise. Let aE + b := az + b : z ∈ E, with a, b fixed complex constants. Prove thatτ(aE + b) = |a|τ(E) for any compact set E ⊂ C.

Exercise. Show that the closed set E = 0 ∪ 1/k : k = 1, 2, . . . has transfinite diameterzero.

Remark. The transfinite diameter τ (considered as a set function) has some of the propertiesof Lebesgue measure on compact subsets of C; in fact, if E is the closed interval [a, b], thenτ([a, b]) = (b−a)/4. However, τ fails to be subadditive; τ(E1∪E2) may exceed the sum τ(E1)+τ(E2).

To investigate the asymptotic behavior of a sequence of Fekete sets Fn, n = 2, 3, . . ., we utilizeweak-star convergence of measures.

Definition 1.2. Let µn be a sequence of finite positive measures with supports‡ supp(µn) ⊂ K

for all n, where K is some compact set. We write µn∗→ µ if

limn→∞

∫

f dµn =

∫

f dµ ∀f ∈ C(K). (1.5)

(If µn(K) ≤ M for some constant M and all n (which clearly holds when µ is a finite measure),this is equivalent to pointwise convergence in the dual space of C(K).) The same definition appliesto signed measures and complex measures. In (1.5), we can always take K to be the extendedcomplex plane C; however, knowing a specific compact set K that contains all the supports of theµn’s serves to remind us that the limit measure will also be supported on K.

†If E consists of only finitely many points, then τ (E) = 0. Why?‡Recall that a point z0 belongs to supp(µ) if and only if every open set containing z0 has positive µ-measure.

E. B. Saff 170

For a discrete set consisting of n points of C, say An = z1, . . . , zn, we associate the normal-ized counting measure

ν(An) :=1

n

n∑

k=1

δzk,

where δz is the unit point mass at z.

Example 1.3. If An+1 consists of the n + 1 Chebyshev nodes for [−1, 1]; that is An+1 =cos(kπ/n) : k = 0, 1, . . . , n, then

ν(An)∗→ dx

π√

1 − x2, (1.6)

which is the arcsine distribution on [−1, 1]. The nodes An+1 are the extreme points of theChebyshev polynomials Tn(x) = cos(n arccos x), which are orthogonal on [−1, 1] with respectto the arcsine distribution. Verify (1.6)!

Example 1.4. If An consists of the nth roots of unity, then ν(An)∗→ 1

2π dθ, where dθ isarclength on the unit circle |z| = 1.

Exercise. Let λ be a real irrational number and let An := exp(λkπi) : k = 1, . . . , n. Prove

that ν(An)∗→ 1

2π dθ. What happens if λ is rational?

As we shall see, many of the results of potential theory are formulated for semi-continuousfunctions.

Definition 1.5. A function f : D → (−∞,∞] (f omits the value −∞) is lower semi-continuous(l.s.c.) on the set D ⊂ C if it satisfies any of the following equivalent conditions:

(i) z ∈ D : f(z) > α is open relative to D for every α ∈ R;

(ii) For every z0 ∈ D,f(z0) ≤ lim inf

z→z0

f(z);

(iii) For every compact subset K ⊂ D, there exists an increasing sequence of continuous functionson K with pointwise limit f .

Exercise. Prove that if f is l.s.c. on a compact set K, then f attains its minimum on K.

Important for us is the fact, which follows from property (iii) and the Monotone ConvergenceTheorem, that if f is l.s.c. on a compact set K, then

∫

Kf dµ ≤ lim inf

n→∞

∫

Kf dµn (1.7)

wherever µn∗→ µ and supp(µn) ⊂ K for all n.

Exercise. Prove that if µn∗→ µ, then for any bounded Borel set E,

µ(E) ≤ lim infn→∞

µn(E) ≤ lim supn→∞

µn(E) ≤ µ(E),


where E and E denote, respectively, the interior of E and the closure of E. [Hint: First show thatthe characteristic function of an open set is l.s.c.]

Our goal now is to determine the weak-star limit (if it exists) for the sequence of normalizedcounting measures ν(Fn) in the Fekete points for a given compact set E. For this purpose we studythe continuous analogue of the discrete minimum energy problem (1.2).

Electrostatics Problem for a Capacitor. Place a unit positive charge on a compact set Eso that equilibrium is attained in the sense that energy is minimized. Again it is assumed that therepulsive force between like-charged particles located at points z and t is proportional to 1/|z − t|.

To create a mathematical framework for this problem, we let M(E) denote the collection of allpositive unit Borel measures µ supported on E (so that M(E) contains all possible distributionsof charges placed on E). The logarithmic potential associated with µ is

Uµ(z) :=

∫

log1

|z − t| dµ(t), (1.8)

which is harmonic outside the support supp(µ) of µ and is l.s.c. in C since

Uµ(z) = limM→∞

∫

min

(

M, log1

|z − t|

)

dµ(t).

The energy of such a potential is defined by

I(µ) :=

∫

Uµ dµ =

∫ ∫

log1

|z − t| dµ(t) dµ(z). (1.9)

Thus, the electrostatics problem involves the determination of

VE := infI(µ) : µ ∈ M(E), (1.10)

which is called the Robin constant for E. Note that since E is bounded, we have

−∞ < VE ≤ +∞.

First we establish the existence of a measure µE ∈ M(E) for which the “inf” is attained. Forthis purpose we use

Lemma 1.6 (Principle of Descent). Let µn be a sequence of measures in M(E) that con-verges weak-star to some µ ∈ M(E). Then, for all z ∈ C,

Uµ(z) ≤ lim infn→∞

Uµn(z), (1.11)

and, furthermore,I(µ) ≤ lim inf

n→∞I(µn). (1.12)

Proof. Inequality (1.11) follows from (1.7) on observing that log 1/|z − t| is l.s.c. in t. Inequality(1.12) follows similarly, on observing that µn × µn converges weak-star to µ× µ.

E. B. Saff 172

Lemma 1.7. There is some µE ∈ M(E) such that I(µE) = VE .

Proof. By the Banach-Alaoglu Theorem, M(E) is compact in the weak-star topology (thisfact is also known as Helly’s Selection Theorem). Let µn be a sequence in M(E) satisfy-ing limn→∞ I(µn) = VE and let ν denote some weak-star cluster point of the µn. Then, by thePrinciple of Descent and the definition of VE , we obtain VE = I(ν).

When VE = +∞ (for example this is the case if E is countable), then every measure µ ∈ M(E)is a minimizing measure. However, if VE is finite, it follows from the strict convexity of I(µ) onM(E) (cf. [ST]) that there exists a unique measure µE such that VE = I(µE). In this case, we callµE the equilibrium measure for E, and UµE the equilibrium or conductor potential for E.

Definition 1.8. The logarithmic capacity of E, denoted by cap(E), is defined by

cap(E) := e−VE . (1.13)

If VE = +∞, we set cap(E) = 0; such sets E are called polar sets because they correspond tosets where potentials can equal +∞. More generally, an arbitrary set E ⊂ C is said to be polar ifevery closed subset of E is polar. In electrostatic terms, polar sets are “too small” to hold a charge.§

Exercise. Prove that any countable set E is a polar set.

Next we establish the connection with the transfinite diameter.

Theorem 1.9. For any compact set E ⊂ C,

τ(E) = cap(E). (1.14)

Moreover, if E has positive capacity, then

ν(Fn)∗→ µE as n→ ∞. (1.15)

Proof. First we show that

VE = log1

cap(E)≥ log

1

τ(E). (1.16)

Let F (z1, z2, . . . , zn) :=∑

1≤i<j≤n log(1/|zi − zj|). Then the expected value of F with respect tothe product of equilibrium measures dµE(z1) dµE(z2) · · · dµE(zn) cannot be less than its minimumvalue defined in (1.2); i.e.,

∫ ∫

· · ·∫

F (z1, z2, . . . , zn) dµE(z1) · · · dµE(zn) =n(n− 1)

2VE

≥ En(E) =n(n− 1)

2log

1

δn(E).

§Somewhat surprising is the fact that the classical “1/3 Cantor set,” which has 1-dimensional Lebesgue measurezero, has positive capacity. The precise value of this capacity is as yet still unknown (see [R2] for some numericalapproximation methods).


Dividing by n(n− 1)/2 and letting n→ ∞ gives (1.16).For the reverse direction, let µ be a weak-star limit point of the measures νn := ν(Fn), say

νn∗→ µ as n→ ∞, n ∈ N . Set logM x := minlog x,M. By the Monotone Convergence Theorem

and the weak-star convergence of νn × νn to µ× µ for n ∈ N , we get

I(µ) =

∫ ∫

log1

|z − t| dµ(z) dµ(t) = limM→∞

∫ ∫

logM

1

|z − t| dµ(z) dµ(t)

= limM→∞

limn→∞

∫ ∫

logM1

|z − t| dνn(z) dνn(t)

≤ limM→∞

limn→∞

2

n2En(E) +

nM

n2

= limM→∞

limn→∞

log1

δn(E)= log

1

τ(E)≤ VE.

Thus from the minimality property of VE, we have

VE ≤ I(µ) ≤ log1

τ(E)≤ VE ,

which proves (1.14). Furthermore, if cap(E) > 0, then by uniqueness of the equilibrium (minimiz-ing) measure, µ = µE . Since µ was an arbitrary limit measure of ν(Fn), (1.15) follows.

As we have earlier observed, Fekete points necessarily lie on the outer boundary of E. Thus from(1.15) we immediately deduce that the equilibrium measure µE is supported on the outer boundaryof E; consequently,

cap(E) = cap(∂∞E), µE = µ∂∞E .

If ∂∞E is a continuum (not a single point), then supp(µE) = ∂∞E. In general, ∂∞E \ supp(µE)has capacity zero.

From our knowledge of Fekete points for the disk we deduce the following.

Example 1.10. If E is the closed disk |z − a| ≤ r, then cap(E) = r and dµE = 12πr ds,¶

where ds is arclength on the circumference |z − a| = r. Furthermore, the equilibrium potentialUµE (z) satisfies

UµE (z) =1

2π

∫ 2π

0log

1

|z − a− reiθ| dθ =

log 1r for |z − a| ≤ r

log 1|z−a| for |z − a| ≥ r.

(1.17)

Exercise. Verify formula (1.17). [Hint: The mean-value property for harmonic functions is usefulhere; see Theorem 2.1.]

¶This also follows from the fact that the disk E is invariant under rotations about z = a and since the equilibriummeasure is unique and supported on the circumference it must also be rotation invariant and hence of the formdescribed.

E. B. Saff 174

Figure 1: Graph of equilibrium potential for the unit disk

Example 1.11. Let E = [a, b] be a segment on the real line. Then cap(E) = (b − a)/4 anddµE is the arcsine measure; i.e.,

dµE =1

π

dx√

(x− a)(b− x), x ∈ [a, b].

If a = −1, b = 1, the conductor potential is given by

UµE (z) = log 2 if x ∈ [−1, 1],

UµE (z) = log 2 − log∣

∣

∣z +√

z2 − 1∣

∣

∣ if z 6∈ [−1, 1],

where the branch√z2 − 1 is positive for z = x > 1. These facts can be obtained from Example

1.10 by applying the Joukowski conformal map of C \ [−1, 1] onto |w| > 1. See Example 3.6.

In the examples for the disk and line segment, observe that the equilibrium potential UµE isconstant on E, namely it equals VE = log 1

cap(E) there. This is certainly consistent with our ex-

pectations based on physical grounds, that equilibrium should occur when the potential (voltage)is constant; for otherwise there would be a flow of charge to the points of E at lower potential.From a mathematically rigorous point of view, this assertion is true quasi-everywhere (q.e.) onE; that is, except for a set of capacity zero.‖ This fact is included in the following result.

Theorem 1.12 (Frostman’s Theorem). Let E ⊂ C be compact with cap(E) > 0. Then

(a) UµE (z) ≤ VE for all z ∈ C;

(b) UµE (z) = VE q.e. on E.

The proof relies on a maximum principle for potentials which is discussed in Section 2. For fulldetails, see [R1], [ST], [Ts].

The theorem suggests that we visualize the 3-dimensional graph of an equilibrium potential assomething like an infinite tent with an “essentially” flat roof consisting of the projection of the setE and tent sides that flow down and outward to −∞; see, e.g., Figure 1.

There are many important consequences of Frostman’s result, of which the following will beuseful in the proof of the main theorem of this section.

‖An arbitrary Borel set B has capacity zero if supcap(F ) : F ⊂ B compact = 0.


Proposition 1.13. Let E ⊂ C be compact with cap(E) > 0. If σ is any probability mea-sure with compact support, then

infz∈E

Uσ(z) ≤ VE = log1

cap(E).

Proof. Here we use the reciprocity law (a simple consequence of the Fubini-Tonelli Theorem)which asserts that

∫

Uσ(z) dµE(z) =

∫

UµE (z) dσ(z).

The left-hand side is bounded below by infz∈E Uσ(z) and, from Frostman’s theorem, VE is an upper

bound for the right-hand side.

We now introduce a third quantity associated with a compact set E — the Chebyshev con-stant, cheb(E) — which arises in a min-max problem.

Polynomial Extremal Problem: Determine the minimal sup norm on E for monic polyno-mials of degree n. That is, determine∗∗

tn(E) := minp∈Pn−1

‖zn + p(z)‖E ,

where Pn−1 denotes the collection of all polynomials of degree ≤ n− 1 and ‖ · ‖E is the sup norm(uniform norm) on E. We assume that E contains infinitely many points (which is always the caseif cap(E) > 0). Then for every n there is a unique monic polynomial Tn(z) = zn + · · · such that‖Tn‖E = tn(E), which is called the nth Chebyshev polynomial for E.

Exercise. Prove that all the zeros of Tn lie in the convex hull of E. (This fact is due to Fejer.)

In view of the simple chain of inequalities

tm+n(E) = ‖Tm+n‖E ≤ ‖TmTn‖E ≤ ‖Tm‖E‖Tn‖E = tm(E)tn(E),

the sequence log tn(E) is subadditive, from which it follows that tn(E)1/n converges and its limitis infk≥1tk(E)1/k (cf. [Ts], [ST]). We call this limit the Chebyshev constant for E:

cheb(E) := limn→∞

tn(E)1/n = infk≥1

tk(E)1/k. (1.18)

From (1.18) and the definition of tn(E) we deduce the following.

Lemma 1.14. For any monic polynomial pn(z) of degree n there holds

‖pn‖E ≥ [cheb(E)]n. (1.19)

∗∗This problem is equivalent to finding the polynomial of degree ≤ n − 1 of best uniform approximation to themonomial zn on E.

E. B. Saff 176

Example 1.15. Let E be the closed disk of radius R, centered at 0. For any p ∈ Pn−1, theratio (zn + p(z))/zn represents an analytic function in |z| ≥ 1 that takes the value 1 at ∞. By themaximum principle for analytic functions,

‖zn + p(z)‖E = max|z|=R

|zn + p(z)| = Rn max|z|=R

∣

∣

∣

∣

zn + p(z)

zn

∣

∣

∣

∣

≥ Rn,

and strict inequality takes place if p(z) is not identically zero. It follows that Tn(z) = zn. Thereforetn(E) = Rn and cheb(E) = R.

Example 1.16. Let E = [−1, 1]. Then Tn is the classical monic Chebyshev polynomial††

Tn(x) = 21−n cos(n arccos x), x ∈ [−1, 1], n ≥ 1.

Thus, tn(E) = ‖Tn‖[−1,1] = 21−n from which it follows that cheb(E) = 1/2, which is the same asthe capacity of [−1, 1].

Exercise. Verify Example 1.16 by using the fact that 21−n cos(n arccos x) equioscillates n + 1times on [−1, 1].

Closely related to Chebyshev polynomials are Fekete polynomials. An nth degree Fekete poly-nomial Fn(z) is a monic polynomial having all its zeros at the n points of a Fekete set Fn.

Example 1.17. If E is the closed unit disk centered at 0, then one can take Fn(z) = zn − 1,so that ‖Fn‖E = 2. Comparing this with Example 1.15 we see that the Fn’s are asymptoticallyoptimal for the Chebyshev problem:

limn→∞

‖Fn‖1/nE = lim

n→∞‖Tn‖1/n

E = 1 = cheb(E).

Moreover, uniformly on compact subsets of |z| > 1, we have

limn→∞

|Fn(z)|1/n = limn→∞

|Tn(z)|1/n = |z| = exp(−UµE (z)),

(the last equality follows from formula (1.17)).

The above examples illustrate the following fundamental theorem, various parts of which aredue to Fekete, Frostman, and Szego.

Theorem 1.18 (Fundamental Theorem of Classical Potential Theory). For any com-pact set E ⊂ C,(a) cap(E) = τ(E) = cheb(E);(b) Fekete polynomials are asymptotically optimal for the Chebyshev problem:

limn→∞

‖Fn‖1/nE = cheb(E) = cap(E).

††There is ambiguity in this notation since Tn(x) is traditionally used to denote cos(n arccos x).


If cap(E) > 0 (so that µE is unique), then we also have:

(c) Fekete points (the zeros of Fn) have asymptotic distribution µE , i.e., ν(Fn)∗→ µE as n→ ∞;

(d) uniformly on compact subsets of the unbounded component of C \ E,

limn→∞

|Fn(z)|1/n = exp(−UµE (z)).

Proof. Part (c) was established in (1.15) of Theorem 1.9. Assertion (d) follows from (c) onobserving that

1

nlog

1

|Fn(z)| = Uν(Fn)(z),

and that all the Fekete points lie on E. Regarding (a) and (b), we already know that cap(E) = τ(E).So to establish (a), we prove that τ(E) = cheb(E).

Let Fn = z(n)k n

k=1 denote an n-point Fekete set for E. Then

δn(n+1)/2n+1 = max

zi⊂E

∏

1≤i<j≤n+1

|zi − zj| ≥[

n∏

k=1

|z − z(n)k |

]

δn(n−1)/2n

for all z ∈ E. Thusδn(n+1)/2n+1 /δn(n−1)/2

n ≥ |Fn(z)|, z ∈ E,

and so on taking nth roots, we get

δ(n+1)/2n+1 /δ(n−1)/2

n ≥ ‖Fn‖1/nE . (1.20)

The left-hand side of this inequality can be written as

(

δn+1

δn

)n/2

δ1/2n+1δ

1/2n ,

which is bounded above by δ1/2n+1δ

1/2n since the sequence δn is decreasing. From (1.19), we have that

the right-hand side of (1.20) is bounded below by cheb(E). Hence

δ1/2n+1δ

1/2n ≥ ‖Fn‖1/n

E ≥ cheb(E),

and so on letting n→ ∞, we get

τ(E) ≥ lim supn→∞

‖Fn‖1/nE ≥ lim inf

n→∞‖Fn‖1/n

E ≥ cheb(E).

It remains only to prove that cheb(E) ≥ τ(E). This is obvious if τ(E) = cap(E) = 0. So assumecap(E) > 0. Let ν(Tn) denote the normalized counting measure in the zeros of Tn(z). Then byProposition 1.13,

infz∈E

Uν(Tn)(z) = infz∈E

1

nlog

1

|Tn(z)| =1

nlog

1

tn(E)≤ VE.

Hencetn(E)1/n ≥ e−VE = cap(E) = τ(E),

E. B. Saff 178

and on letting n→ ∞, we get cheb(E) ≥ τ(E).

Exercise. Let 0 < a < b. Prove that

cap ([a, b] ∪ [−b,−a]) =

√b2 − a2

2.

[Hint: What are the Chebyshev polynomials of even degree for this union?]

Exercise. Let P (z) be a monic polynomial of degree n, and consider the lemniscate set L :=z : |P (z)| ≤ Rn. Prove that cap(L) = R. [Hint: Begin by determining the Chebyshev polynomi-als Tkn, k = 1, 2, . . . , for L.]

Exercise. Let E be a compact set and ǫ > 0. Show that there exists a lemniscate set L such thatE ⊂ L and cap(L) < cap(E) + ǫ.

2 Harmonic, Superharmonic and Subharmonic Functions

Recall that a real-valued function u(z) defined in an open set D ⊂ C is harmonic in D if u andits 1st and 2nd partial derivatives are continuous in D and u satisfies Laplace’s equation

uxx(z) + uyy(z) = 0, z ∈ D. (2.1)

(Actually it is enough to merely assume that the 2nd partial derivatives exist and satisfy (2.1).)Locally, harmonic functions are the real (or imaginary) parts of an analytic function.

Exercise. Prove that if u is harmonic in D, then g(z) := ux(z) − iuy(z) is analytic in D.

Note that the important function log |z| is harmonic in C \ 0, is locally the real part of abranch of log z, but is not globally (in C \ 0) the real part of an analytic function.

Exercise. Prove that the logarithmic potential Uµ(z) is harmonic for z not in the support ofµ (assuming this support is compact and µ is a finite measure).

Harmonic functions can also be characterized by the following Mean-Value Property (MVP).

Theorem 2.1. A real-valued function u(z) is harmonic in an open set D if and only if u iscontinuous in D and locally satisfies the mean-value property; i.e., if the disk |z − a| ≤ r iscontained in D, then

u(a) =1

2π

∫ 2π

0u(a+ reiθ) dθ. (2.2)

(In fact, it is enough that equality holds for r = r(a) sufficiently small.)

Exercise. Prove that if u is harmonic on an open set D containing the disk |z − z0| ≤ r, then

u(z0) =1

πr2

∫ ∫

|z−z0|≤ru(z) dxdy.


Exercise. Prove that if un(z), n = 1, 2, . . ., is a sequence of functions harmonic in D that convergeslocally uniformly to a function u in D, then u is harmonic in D.

An important consequence of (2.2) is the Max-Min Principle for harmonic functions.

Theorem 2.2. If u is harmonic in a domain D (i.e., an open connected set) and u attainsits maximum (or minimum) in D, then u is identically constant in D.

Furthermore, if u is harmonic in the interior and continuous on the boundary of a compact set,then u attains its max and min on the boundary.

The proof of this principle follows from the observation that if u attains its max at a pointz0 ∈ D and Dr(z0) is any closed disk centered at z0, then u must equal u(z0) for all z ∈ Dr(z0)since the contrary assumption would lead to a violation of the MVP (simply integrate around acircle centered at z0 and containing a point where u(z) < u(z0)).

Theorem 2.2 also tells us that harmonic functions are determined by their values on the bound-ary of a compact set. Indeed, in the case of a disk, we have Poisson’s integral formula: If u isharmonic in |z| < R and continuous on |z| ≤ R, then

u(z) =1

2π

∫ 2π

0P (t, z)u(t) dθ, t = Reiθ, |z| < R, (2.3)

where

P (t, z) :=|t|2 − |z|2|t− z|2 = ℜ

(

t+ z

t− z

)

. (2.4)

This formula can be deduced, for example, from the Cauchy integral formula for analytic func-tions; it includes as a special case the MVP (2.2).

Exercise. Prove that if U(t) is integrable (in the Lebesgue sense) on |t| = R, then

u(z) :=1

2π

∫ 2π

0P (t, z)U(t) dt (2.5)

is harmonic in |z| < R. (If U(t) is continuous on the circle |t| = R, then Schwarz’s theorem assertsthat u as given in (2.5) solves the Dirichlet problem for the disk; i.e., limz→t u(z) = U(t) for all ton the boundary |t| = R.)

If we replace equality in (2.2) by ≥, we obtain the class of superharmonic functions.

Definition 2.3. An extended real-valued function f on an open set D ⊂ C is calledsuperharmonic in D if f is not the constant function +∞ and satisfies

(i) f is lower semi-continuous on D;

(ii) the value of f at any point z0 ∈ D is not less than its average over any circle in D centeredat z0; that is

f(z0) ≥1

2π

∫ 2π

0f(z0 + reiθ) dθ (2.6)

E. B. Saff 180

provided the closed disk Dr(z0) := z ∈ C : |z − z0| ≤ r is contained in D.

A subharmonic function is the negative of a superharmonic function. A real-valued functionu(z) that is both superharmonic and subharmonic in D is harmonic in D.

Exercise. Let f : D → R, D a domain, f ∈ C2(D). Prove that f is superharmonic in D ifand only if ∆f := fxx + fyy ≤ 0 at all points of D. [Hint: Begin by showing that a negativeLaplacian implies that f is superharmonic.]

As the above exercise illustrates, superharmonic functions in R2 are analogues of concave func-

tions in R; and subharmonic functions are the analogues of convex functions.

Exercise. Prove that if F (z) is analytic in a domain D and p > 0, then |F (z)|p is subhar-monic in D. Furthermore, show log |F (z)| is subharmonic in D unless F is identically zero.

Essential for us is the fact that logarithmic potentials are superharmonic in C. Indeed, as wehave seen,

Uµ(z) =

∫

log1

|z − t| dµ(t)

is l.s.c. on C and since log(1/|z − t|) is superharmonic in C for fixed t (Why?), it follows from theFubini-Tonelli theorem that for a ∈ C

1

2π

∫ 2π

0Uµ(a+ reiθ) dθ =

∫

1

2π

∫ π

−πlog

1

|a+ reiθ − t| dθ dµ(t)

≤∫

log1

|a− t| dµ(t) = Uµ(a)

(see also (1.17)).

Exercise. Prove that if Uµ is harmonic in a neighborhood of a point z0, then z0 6∈ supp(µ).

While it may appear that potentials are rather special types of superharmonic functions, theirproperties are key to the analysis of general superharmonic functions. This is thanks to the follow-ing celebrated result.

Theorem 2.4 (Riesz Decomposition). If f is superharmonic in a domain D, then there existsa positive measure λ supported on D such that for every subdomain D∗ ⊂ D for which D∗ ⊂ D,we have

f(z) = h(z) +

∫

Dlog

1

|z − t| dλ(t), z ∈ D∗, (2.7)

where h is harmonic in D∗.

For the case when f is smooth, superharmonicity implies that ∆f = fxx + fyy ≤ 0 in D and itturns out that the positive measure

dλ(t) := − 1

2π∆f(t) dm2(t), t ∈ D, (2.8)


where m2 denotes 2-dimensional Lebesgue measure, yields the appropriate potential Uλ for which(2.7) holds.

Let’s verify this for the simple but important case when f(z) = −|z|2, for which we have∆f ≡ −4. Then we need to show that ∆(f − Uλ) = 0 in D∗ where dλ = (2/π) dm2. Foran arbitrary disk Dr(z0) := z : |z − z0| < r contained in D∗, write Uλ = Uλ1 + Uλ2 whereλ1 = λ|Dr(z0) and λ2 = λ − λ1. Then Uλ2 is harmonic in Dr(z0) and so we need only show that

∆(f − Uλ1) = 0 in Dr(z0). A simple calculation using (1.17) gives that

Uλ1(z) =2

π

∫

Dr(z0)log

1

|z − t| dm2(t)

= 2r2 log1

r+ r2 − |z − z0|2,

from which we get ∆Uλ1(z) = −4 = ∆f for z ∈ Dr(z0), as desired. For more general but smoothf , one can use Green’s formula‡‡ to verify that (2.8) yields the decomposition in (2.7).

For general superharmonic functions f , we interpret the right-hand side of (2.8) in the distribu-tional sense (cf. [R1, Sec. 3.7]); more precisely, we identify −∆f dm2 as the unique positive measurethat satisfies

∫

Dφ(−∆f dm2) := −

∫

Df∆φdm2 (2.9)

for all C∞ functions φ whose support is a compact subset of D. This condition is precisely whatwould be expected from Green’s formula. The existence of such a measure −∆f dm2 satisfying (2.9)is guaranteed by the Riesz representation theorem for linear functionals. With this interpretation,it follows that for any finite Borel measure µ with compact support there holds

µ = − 1

2π∆Uµ. (2.10)

Just as the MVP (2.2) for harmonic functions implied the Max-Min Principle (Theorem 2.2),the mean-value inequality property (2.6) yields the following.

Theorem 2.5 (Minimum Principle for Superharmonic Functions). Let D be a boundeddomain and g a superharmonic function on D such that

lim infz→ζ

g(z) ≥ m ∀ ζ ∈ ∂D. (2.11)

Then g(z) > m for all z in D, unless g is constant.

Exercise. Use the Min Principle to prove that a l.s.c. function f (not identically +∞) is su-perharmonic in a domain D if and only if it has the following property: If D0 ⊂ D is a boundeddomain whose closure is contained in D and u is harmonic in D0, continuous on D0 and satisfies

‡‡Recall that Green’s formula for smooth functions u, v on a bounded open set D with smooth boundary ∂Dasserts that

∫

D

(v∆u − u∆v) dm2 = −

∫

∂D

(v∂u

∂n− u

∂v

∂n) ds,

where ∂/∂n denotes differentiation in the direction of the inner normal to D.

E. B. Saff 182

u(ζ) ≤ f(ζ) ∀ ζ ∈ ∂D0, then u(z) ≤ f(z) ∀ z ∈ D0.

A more general form of the Min Principle allows us to ignore a set of points on ∂D of capacityzero provided g is lower bounded on D; see [ST].

Theorem 2.6 (Generalized Min Principle). If D ⊂ C is a domain, cap(∂D) > 0, g is super-harmonic and bounded from below in D and (2.11) holds for q.e. ζ on ∂D, then g(z) > m ∀ z ∈ Dunless g is constant.

Exercise. Give an example to show that the lower boundedness assumption cannot be removedin the above result.

For potentials what is crucial is their behavior on the support of its defining measure.

Theorem 2.7 (Maximum Principle for Potentials). Let µ be a finite positive measure withcompact support. If Uµ(z) ≤M for all z ∈ supp(µ), then Uµ(z) ≤M for all z ∈ C.

The Max Principle is a special case of the following important result.

Theorem 2.8 (Principle of Domination). Let µ, ν be positive finite measures with com-pact supports, ν(C) ≤ µ(C), and µ has finite logarithmic energy (I(µ) <∞). If for some constantc, the inequality

Uµ(z) ≤ Uν(z) + c

holds µ-a.e., then it holds for all z ∈ C.

The idea of the proof of the above theorem is to consider the function

U(z) := min(Uν(z) + c, Uµ(z)),

which is superharmonic since the minimum of two superharmonic functions is again superharmonic.Let λ := − 1

2π∆U (i.e., λ is the measure guaranteed by the Riesz Decomposition Theorem (RDT)with D = C) and argue that λ must equal µ. See [ST] for details.

As an application of the Principle of Domination, we present

Example 2.9. Let pn, n = 1, 2, . . ., be a sequence of monic polynomials of respective degrees nthat satisfy

lim supn→∞

‖pn‖1/n[−1,1] ≤ 1

2= cap([−1, 1]), (2.12)

and let ν(pn) denote the normalized counting measure in the zeros of pn. If all the zeros of the pn’slie on [−1, 1], then

ν(pn)∗→ µ[−1,1] =

dx

π√

1 − x2as n→ ∞. (2.13)

Indeed, by definition of the sup norm,

1

nlog

1

|pn(z)| ≥1

nlog

1

‖pn‖[−1,1], z ∈ [−1, 1].


We write this last inequality in the equivalent form

Uν(pn)(z) + log 2 ≥ 1

nlog

1

‖pn‖[−1,1]+ Uµ[−1,1](z), z ∈ [−1, 1], (2.14)

where we used the fact that Uµ[−1,1](x) = log 2 for all x ∈ [−1, 1] (see Examples 1.11 and 3.6). Bythe Principle of Domination, (2.14) holds for all z ∈ C. Now let ν be a weak-star limit measure ofthe sequence ν(pn). Then from (2.12) and (2.14), we get

Uν(z) + log 2 ≥ log 2 + Uµ[−1,1](z) ∀ z 6∈ [−1, 1],

i.e., Uν(z) − Uµ[−1,1](z) ≥ 0 for all z ∈ Ω := C \ [−1, 1]. But since Uν − Uµ[−1,1] vanishes at ∞ andis harmonic in Ω, the Max-Min Principle asserts that Uν(z) = Uµ[−1,1](z) for z ∈ Ω. By l.s.c., wethen deduce that Uν(x) ≤ Uµ[−1,1](x) = log 2 for all x ∈ [−1, 1] and so I(ν) ≤ log 2 = I(µ[−1,1]).By uniqueness of the minimizing measure, we get that ν = µ[−1,1], which proves (2.13).

Actually, (2.13) holds without any prior assumptions on the location of zeros of the pn’s. Thisfollows from the fact that (2.12) implies that the proportion of zeros that lie outside any open setcontaining [−1, 1] is asymptotically negligible (see Section 3).

Another consequence of the Riesz Decomposition Theorem is the following.

Theorem 2.10 (Unicity Theorem). Let µ, ν be positive finite measures having compactsupport. If, in a region D ⊂ C, there holds

Uµ(z) = Uν(z) + h(z) m2-a.e.,

where h is harmonic in D, then µ|D = ν|D.

In particular, if two potentials Uµ and Uν agree except for a set of 2-dimensional Lebesguemeasure zero, then µ = ν.

3 Equilibrium Potentials, Green Functions and Regularity

Throughout this section, E ⊂ C denotes a compact set with cap(E) > 0. Here, we discuss propertiesand characterizations of the equilibrium (conductor) potential UµE .

According to Frostman’s Theorem 1.12, UµE is “essentially” constant on E (more precisely,constant q.e. on E). So the following result should come as no surprise.

Theorem 3.1. If ν ∈ M(E) has finite logarithmic energy (i.e., I(ν) < ∞) and Uν(z) = cq.e. on E, then c = VE and ν = µE.

In the proof of this result, a useful fact is the following.

Exercise. If a measure ν has finite logarithmic energy, then any set of capacity zero has ν measurezero.

With this fact, Theorem 3.1 follows by simply integrating the equality Uν = c with respect todµE and interchanging order of integration.

E. B. Saff 184

Exercise. Give an example to show that if ν ∈ M(E) and Uν is constant on E except forone point, then ν need not equal µE.

What can be said about the continuity properties of UµE? Certainly UµE is continuous inC \ supp(µE) since it is harmonic there. So suppose z0 ∈ supp(µE). Then by l.s.c. and Frostman’sTheorem, we have

UµE (z0) ≤ lim infz→z0

UµE (z) ≤ lim supz→z0

UµE (z) ≤ VE.

Hence if UµE (z0) = VE , then UµE is continuous at z0. The converse is also true. (Prove it!) Tosummarize, we have:

Theorem 3.2. UµE is continuous at z0 ∈ supp(µE) if and only if UµE (z0) = VE. Conse-quently, UµE is continuous q.e. in the plane.

Definition 3.3. A point z0 ∈ ∂∞E is said to be a regular point of the unbounded com-ponent Ω of C \ E if UµE (z0) = VE . Otherwise, z0 is called an irregular point. (From Theorem3.2, we see that the set of all irregular points has capacity zero.) If every point of ∂Ω = ∂∞E isregular, we say that Ω is regular (with respect to the Dirichlet problem).

Exercise. Prove that every interior point of E satisfies UµE (z) = VE .

The equilibrium potential is related to the Green function associated with the unbounded com-ponent of the complement of E; more precisely, we have

Definition 3.4. The Green function with pole at ∞ for the unbounded component Ωof C \ E is defined by

gΩ(z,∞) := VE − UµE (z). (3.1)

(Some authors write gE(z,∞) instead of gΩ(z,∞).)

Three properties uniquely characterize this function for z ∈ Ω; namely

(a) gΩ(z,∞) is harmonic in Ω\∞ and bounded from above and below outside each neighborhoodof ∞;

(b) gΩ(z,∞) − log |z| = O(1) as z → ∞;

(c) gΩ(z,∞) → 0 as z → ζ, z ∈ Ω, for q.e. ζ ∈ ∂Ω.

Regarding property (b), it follows from (3.1) and the fact that µE is a unit measure that

gΩ(z,∞) − log |z| → VE = log1

cap(E)as z → ∞. (3.2)

It is also clear from (3.1) that gΩ(z,∞) ≥ 0, gΩ(z,∞) > 0 for z ∈ Ω, and in view of Theorem 3.2,if ζ ∈ ∂Ω = ∂∞E, then gΩ(z,∞) → 0 as z → ζ, z ∈ Ω, if and only if ζ is a regular point of Ω.


Exercise. Prove that if g(z) is a function that satisfies properties (a), (b), and (c) for z ∈ Ω,then g(z) = gΩ(z,∞) for z ∈ Ω.

In the case when Ω is simply connected, we can relate the Green function to a Riemann mappingof Ω.

Theorem 3.5. If the unbounded component Ω of C \ E is simply connected, then gΩ(z,∞) =log |Φ(z)|, z ∈ Ω, where w = Φ(z) is the unique Riemann mapping function from Ω to the exteriorw : |w| > 1 of the unit disk such that Φ(∞) = ∞, Φ′(∞) > 0.

Such a function Φ has a Laurent expansion about ∞ of the form

Φ(z) =z

c+ a0 +

a1

z+a2

z2+ · · · , with c > 0, (3.3)

and using this representation, properties (a), (b), and (c) are easy to establish.

Exercise. Prove Theorem 3.5.

From (3.3), we immediately see that

log |Φ(z)| − log |z| → log1

cas z → ∞,

and comparison with (3.2) shows that

c = cap(E).

From this fact, we can determine the capacity of any compact set E providing we know the exteriorconformal mapping function Φ. We illustrate this for the line segment.

Example 3.6. Let E = [−1, 1]. The well-known Joukowski transformation z = ψ(w) =12(w+w−1) maps the exterior of the unit circle onto Ω := C \ [−1, 1], with ψ(∞) = ∞, ψ′(∞) > 0.Solving for w, we obtain the desired Riemann mapping:

w = Φ(z) = z +√

z2 − 1,

where√z2 − 1 behaves like z near infinity. Thus

gΩ(z,∞) = log∣

∣

∣z +√

z2 − 1∣

∣

∣ ,

and since Φ(z) = 2z + · · · near infinity, we get that cap([−1, 1]) = 1/2. Furthermore, from (3.1),we have

UµE (z) = log 2 − log∣

∣

∣z +

√

z2 − 1∣

∣

∣

as claimed in Example 1.11.

Exercise. Show that if the unbounded component Ω of C \ E is simply connected, then ev-ery point of ∂∞E is regular, i.e., Ω is a regular domain.

E. B. Saff 186

Exercise. By constructing a suitable mapping function, show that the capacity of an ellipsewith semi-axis lengths a and b is (a+ b)/2.

Exercise. Show that if the compact set E has positive capacity and p(z) is a monic polyno-mial of degree n, then the set p−1(E) has capacity [cap(E)]1/n.

In the case when ∂∞E is a smooth closed Jordan curve, there is a simple representation for µE

in terms of the Green function gΩ = gΩ(z,∞). Using Green’s formula, one first shows that, forz ∈ Ω, the equilibrium potential (= VE − gΩ) identically equals the potential of the unit measure12π

∂gΩ∂n ds, where the derivative is taken in the direction of the outer normal on ∂∞E and s denotes

arclength on ∂∞E (cf. [W, Sec. 4.2 ]). On letting z ∈ Ω approach ∂∞E and appealing to the lowersemi-continuity of potentials, it follows that 1

2π∂gΩ∂n ds has energy at most VE, and so by uniqueness

of the minimizing measure, we deduce that dµE = 12π

∂gΩ∂n ds. Thus, for any Borel subset γ of ∂∞E,

µE(γ) =1

2π

∫

γ

∂gΩ∂n

ds =1

2π

∫

γ|Φ′|ds.

Alternatively, µE(γ) is given by the normalized angular measure of the image Φ(γ):

µE(γ) =1

2π

∫

Φ(γ)dθ (3.4)

(for this representation, the smoothness of ∂∞E is not needed).The Green function is especially useful for estimating the modulus of a polynomial outside E

when its sup norm on E is known.

Lemma 3.7. (Bernstein-Walsh). If pn(z) is any polynomial of degree ≤ n, then

|pn(z)| ≤ ‖pn‖E engΩ(z,∞), z ∈ Ω, (3.5)

where ‖pn‖E := maxz∈E |pn(z)| and Ω is the unbounded component of C \ E.

Proof. Assume that pn has exact degree n. Then (3.5) is equivalent to

1

nlog

1

|pn(z)| + gΩ(z,∞) ≥ 1

nlog

1

‖pn‖E, z ∈ Ω. (3.6)

Let u(z) denote the left-hand side of (3.6) and note that u is superharmonic in Ω and harmonic at∞. Moreover, from property (c) for gΩ, we deduce that

lim infz→ζz∈Ω

u(z) ≥ 1

nlog

1

‖pn‖Efor q.e. ζ ∈ ∂Ω.

Thus (3.6) follows from the Minimum Principle for Superharmonic Functions.

It is useful to consider Green functions with poles at finite points of the plane. If D is a domainwith cap(∂D) > 0, ∂D ⊂ C compact, the Green function gD(z, ζ) for D with pole at ζ ∈ D isthe unique real-valued function of z satisfying


(a’) gD(z, ζ) is harmonic in D \ ζ and bounded outside any neighborhood of ζ.

(b’) gD(z, ζ) − log 1|z−ζ| = O(1) as z → ζ.

(c’) limz→wz∈D

gD(z, ζ) = 0 for q.e. w ∈ ∂D.

The relation of this Green function to the one with pole at ∞ is easy to see. Consider themapping w = 1/(z − ζ) that takes ζ to ∞ and D to some domain D′. Then

gD(z, ζ) = gD′

(

1

z − ζ,∞

)

. (3.7)

Exercise. Verify that the function of z on the right-hand side of (3.7) satisfies properties (a’),(b’), (c’).

Exercise. Verify that for the unit disk D : |z| < 1,

gD(z, ζ) = log

∣

∣

∣

∣

1 − ζz

z − ζ

∣

∣

∣

∣

, z, ζ ∈ D.

A clever application of Green’s formula shows that gD is symmetric: gD(z, ζ) = gD(ζ, z); see[Ts].

Exercise. Prove that if pn, n = 1, 2, . . ., is a sequence of monic polynomials of respective de-grees n that satisfy

lim supn→∞

‖pn‖1/n[−1,1] ≤ 1

2= cap([−1, 1]),

then the proportion of the number of zeros of the pn’s that lie outside any neighborhood of [−1, 1]tends to zero as n → ∞. (Recall Example 2.9 and the remark following it.) [Hint: Let zn,kk∈Jn

denote the zeros of pn that lie at a distance ≥ ǫ > 0 from [−1, 1] and consider the functions

1

nlog

1

|pn(z)| −1

n

∑

k∈Jn

g(z, zn,k) + g(z,∞),

where g = gC\[−1,1].]

Just as log 1|z−t| serves as the kernel for logarithmic potential theory, so too does gD(z, t) serve

as the kernel for Green potential theory. If ν is a finite positive measure on D with compact supportin C, we define

UνD(z) :=

∫

gD(z, ζ) dν(ζ), z ∈ D, (3.8)

and note that UνD ≥ 0 in D and Uν

D is superharmonic in D and harmonic in D \ supp(ν). Further-more, if ν has compact support in D, then

limz→wz∈D

UνD(z) = 0 for q.e. w ∈ ∂D.

E. B. Saff 188

The Green energy of a measure ν is defined by

ID(ν) :=

∫ ∫

gD(z, ζ) dν(z) dν(ζ). (3.9)

For a closed subset E ⊂ D of positive logarithmic capacity, we consider the minimum energyproblem

V DE := infID(ν) : ν ∈ M(E) (3.10)

for which there exists a unique measure (the Green equilibrium measure) µDE ∈ M(E) such

that ID(µDE ) = V D

E . Analogous to Frostman’s Theorem 1.12, there holds

UµD

E

D (z) = V DE q.e. on E (3.11)

UµD

E

D (z) ≤ V DE for all z ∈ D. (3.12)

The constant

cap(E, ∂D) :=1

V DE

(3.13)

is called the capacity of the condenser (E, ∂D); see Section 5.

BalayageLet D ⊂ C be an open set with compact boundary ∂D of positive capacity and let µ be a measurewith supp(µ) ⊂ D. The problem of balayage (a French word meaning “sweeping”) consists offinding a new measure µb supported on ∂D such that µb(C) = µ(C) and

Uµb

(z) = Uµ(z) for q.e. z 6∈ D. (3.14)

For a bounded domain D, the sweeping out of the measure µ to ∂D can always be accomplished,but if the domain D ⊂ C contains the point at infinity, it is necessary to modify (3.14) so that itreads

Uµb

(z) = Uµ(z) + c for q.e. z 6∈ D, (3.15)

for some constant c. Necessarily

c =

∫

gD(z,∞) dµ(z). (3.16)

If D is connected and regular, then equality in (3.14) and (3.15) holds for all z 6∈ D. To ensure

uniqueness of µb, a condition such as boundedness of Uµbon ∂D suffices.

Exercise. Verify that the constant c in (3.15) is given by (3.16). [Hint: Starting with∫

gD(z,∞) dµ(z) =

∫

[VE − Uµ∂D(z)] dµ(z),

use the reciprocity law together with (3.15) on ∂D.]

Balayage measures can also be characterized by the following property: if h is any function thatis continuous on D and harmonic in D, then

∫

Dhdµ =

∫

∂Dhdµb. (3.17)


Example 3.8. If D is the unit disk |z| < 1 and δζ is the unit point mass at ζ ∈ D, thenthe balayage of δζ to ∂D : |z| = 1 is given by

dδbζ(t) =

1

2πP (t, ζ) dθ, t = eiθ, 0 ≤ θ ≤ 2π,

where P denotes the Poisson kernel (2.4).

If D is a compact subset of C with cap(D) > 0, then the balayage of δ∞ onto the outer boundaryof D is the equilibrium measure µD.

The notion of balayage is intimately connected to the Green function of a domain. Indeed, ifD is bounded, ζ ∈ D, and δb

ζ denotes the balayage of δζ to ∂∞D, then

gD(z, ζ) = log1

|z − ζ| − U δbζ (z), (3.18)

since, as can be verified, the right-hand satisfies properties (a’), (b’), and (c’) that characterize theGreen function with pole at ζ. If µ is a finite positive measure on D, then

µb =

∫

δbζ dµ(ζ) (3.19)

and so, on integrating (3.18) with respect to dµ(ζ), we get

UµD(z) =

∫

gD(z, ζ) dµ(ζ) = Uµ(z) − Uµb

(z), z ∈ D. (3.20)

As a consequence of (3.20) and the nonnegativity of the Green potential, we get

Uµb

(z) ≤ Uµ(z) ∀z ∈ C. (3.21)

In case D is an unbounded domain with ∂D ⊂ C, then (3.18), (3.19), and (3.20) must bemodified to include the constant c of (3.16); e.g., (3.21) becomes

Uµb

(z) ≤ Uµ(z) + c ∀z ∈ C. (3.22)

4 Applications to Polynomial Approximation of Analytic Functions

Let f be a continuous complex-valued function on a compact set E ⊂ C and let

en(f ;E) = en(f) := minp∈Pn

‖f − p‖E (4.1)

be the error in best uniform approximation of f by polynomials of degree at most n. We denoteby p∗n the polynomial of best approximation: ‖f − p∗n‖E = en(f).

If en(f) → 0 as n→ ∞, the series

p∗1 +

∞∑

n=1

(p∗n+1 − p∗n)

E. B. Saff 190

converges to f uniformly on E, so that the continuous function f must be analytic at every interiorpoint of E. (The collection of all functions that are continuous on E and analytic in the interior ofE is denoted by A(E).) Furthermore, it follows from the maximum principle for analytic functions,that the above series automatically converges on every bounded component of C \ E, so that itssum represents an analytic continuation of f to these components (e.g., if E is the unit circle|z| = 1, then the convergence holds in the unit disk |z| ≤ 1). Such a continuation, however, maybe impossible. Therefore, in order to ensure that en(f) → 0 for every function f in A(E), it isnecessary to assume that the only component of C \ E is the unbounded one; that is, C \ E isconnected (so that E does not separate the plane).

A celebrated theorem of S.N. Mergelyan (cf. [Ga]) asserts that this assumption is also sufficient.Here, we prove this result in a special case when E has a connected and regular complementΩ := C \ E and f is analytic in some neighborhood of E. Our aim is to determine the rate ofapproximation.

For any R > 1, let ΓR denote the level curve z : gΩ(z,∞) = logR, see Fig. 2 (we call such acurve a level curve with index R). The assumption that Ω is regular ensures that for any openset V containing E, the level curve ΓR will lie in V for R sufficiently close to 1.

Φ1

ΓR

E

R

Figure 2: Level curve of gΩ(z,∞)

Let Fn+1 be the (n+ 1)-st degree Fekete polynomial for E and let Pn be the polynomial of degree≤ n that interpolates f at the zeros of Fn+1. We are given that f is analytic in a neighborhood ofE; hence there exists R > 1 such that f is analytic on and inside ΓR. For any such R, the Hermiteinterpolation formula yields

f(z) − Pn(z) =1

2πi

∫

ΓR

Fn+1(z)

Fn+1(t)

f(t) dt

t− z, z inside ΓR. (4.2)

(The validity of the Hermite formula follows by first observing that the right-hand side vanishes atthe zeros of Fn+1(z), and then by replacing f(z) by its Cauchy integral representation to deducethat the difference between f and the right-hand side is indeed a polynomial of degree at most n.)

Formula (4.2) leads to a simple estimate:

en(f) ≤ ‖f − Pn‖E ≤ K‖Fn+1‖E

minΓR|Fn+1(t)|

,

where K is some constant independent of n. Applying parts (b), (c) of the Fundamental Theorem1.18, we obtain that

lim supn→∞

en(f)1/n ≤ cap(E)

R cap(E)=

1

R< 1. (4.3)

We have proved that indeed en(f) → 0 and that the convergence is geometrically fast. Since R > 1was arbitrary (but such that f is analytic on and inside ΓR), we have actually proved that (4.3)


holds with R replaced by R(f), where

R(f) := supR : f admits analytic continuation to the interior of ΓR.Can we improve on this? The answer is — no! In order to show this, we appeal to the Bernstein-Walsh Lemma 3.7.

Assume now that (4.3) holds for some R > R(f) and let R(f) < ρ < R. Then for some constantc > 1,

en(f) ≤ c

ρn, n ≥ 1.

Since, from the triangle inequality,

‖p∗n+1 − p∗n‖E = ‖p∗n+1 − f + f − p∗n‖E ≤ en+1(f) + en(f) ≤ 2cρ−n,

we obtain from the Bernstein-Walsh Lemma that for any r > 1,

‖p∗n+1 − p∗n‖ΓR≤ 2c

(

r

ρ

)n

, n ≥ 1.

If we choose R(f) < r < ρ, we obtain that the series p∗1 +∑∞

n=1(p∗n+1 − p∗n) converges uniformly

inside Γr. Hence it gives an analytic continuation of f to the interior of Γr, which contradicts thedefinition of R(f).

Let us summarize what we have proved.

Theorem 4.1 (Walsh [W, Ch. VII]). Let E be a compact set with connected and regular com-plement. Then for any f ∈ A(E),

lim supn→∞

en(f)1/n =1

R(f).

Remark. R(f) is the first value of R for which the level curve ΓR contains a singularity of f .It may well be possible that f is analytic at some other points of ΓR(f), but the geometric rate ofbest polynomial approximation “does not feel this” — whether every point of ΓR(f) is a singularityor merely one point is a singularity, the rate of approximation remains the same as if f was analyticonly inside of ΓR(f)! To take advantage of any extra analyticity, different approximation tools areneeded; e.g., rational functions.

Example 4.2. Let E = [−1,−α] ∪ [α, 1], 0 < α < 1, and let f = 0 on [−1,−α] and f = 1on [α, 1]. Some level curves ΓR of gC\E are depicted on Fig. 3. For R small, ΓR consists of twopieces, while for R large, ΓR is a single curve. There is a “critical value” R0 = gC\E(0,∞) for whichΓR0 represents a self-intersecting lemniscate-like curve (the bold curve in Fig. 3). Clearly, f can beextended as an analytic function to the interior of ΓR0 (define f = 0 inside the left lobe and f = 1inside the right lobe). For R > R0, the interior of ΓR is a (connected) domain; hence there is nofunction analytic inside of ΓR that is equal to 0 on [−1,−α] and to 1 on [α, 1]. Therefore

R(f) = R0 = exp

gC\E(0,∞)

,

and by Theorem 4.1:lim sup

n→∞en(f)1/n = exp

−gC\E(0,∞)

.

E. B. Saff 192

ΓR, R > R0

−1

ΓR, R < R0

−α 1α0

ΓR0

Figure 3: Level curves of gC\E for Example 4.2

5 Rational Approximation

For a rational function R(z) = P1(z)/P2(z), where P1 and P2 are monic polynomials of degree n,one can write

− 1

nlog |R(z)| = Uν1(z) − Uν2(z),

where ν1, ν2 are the normalized zero counting measures for P1, P2, respectively. The right-handside represents the logarithmic potential of the signed measure µ = ν1 − ν2:

Uν1(z) − Uν2(z) = Uµ(z) =

∫

log1

|z − t| dµ(t).

The theory of such potentials can be developed along the same lines as in the earlier sections.We present below only the very basic notions of this theory that are needed to formulate theapproximation results. A more in-depth treatment can be found in the works of Bagby [B], Gonchar[Gon], as well as [ST].

The analogy with electrostatics problems suggests considering the following energy problem.Let E1, E2 ⊂ C be two closed sets that are a positive distance apart. The pair (E1, E2) is calleda condenser and the sets E1, E2 are called the plates. Let µ1 and µ2 be positive unit measuressupported on E1 and E2, respectively. Consider the energy integral of the signed measure µ =µ1 − µ2:

I(µ) =

∫ ∫

log1

|z − t| dµ(z) dµ(t).

Since µ(C) = 0, the integral is well-defined, even if one of the sets is unbounded. While not obvious,it turns out that such I(µ) is always positive. We assume that E1 and E2 have positive logarithmiccapacity. Then the minimal energy (over all signed measures of the above form)

V (E1, E2) := infµI(µ)

is finite and positive. We then define the condenser capacity cap(E1, E2) by

cap(E1, E2) := 1/V (E1, E2).


One can show, as with the Frostman theorem, that there exists a unique signed measureµ∗ = µ∗1 − µ∗2 (the equilibrium measure for the condenser) for which I(µ∗) = V (E1, E2). Fur-thermore, the corresponding potential (called the condenser potential) is constant on each plate:

Uµ∗= c1 on E1, Uµ∗

= −c2 on E2 (5.1)

(we assume throughout that E1, E2 are regular — otherwise the above equalities hold only quasi-everywhere). On integrating against µ∗, we deduce from (5.1) that

c1 + c2 = V (E1, E2) = 1/cap(E1, E2). (5.2)

We mention that (similar to the case of the conductor potential) the relations of type (5.1) charac-terize µ∗. Moreover, one can deduce from (5.1) that the measure µ∗i is supported on the boundary(not necessarily the outer one) of Ei, i = 1, 2. Therefore, on replacing each Ei by its boundary, wedo not change the condenser capacity or the condenser potential.

Example 5.1. Let E1, E2 be, respectively, the circles |z| = r1 |z| = r2, r1 < r2. Thesesets are invariant under rotations. Being unique, the measure µ∗ is therefore also invariant underrotations and we obtain that

dµ∗1 =1

2πr1ds, dµ∗2 =

1

2πr2ds,

where ds denotes the arclength over the respective circles E1, E2. Applying the result of Example1.10, we find that

Uµ∗(z) =

0, |z| > r2log(r2/|z|), r1 ≤ |z| ≤ r2log(r2/r1), |z| < r1.

Therefore (recall (5.2))

cap(E1, E2) = 1/ logr2r1. (5.3)

Assume now that each plate of a condenser is a single Jordan arc or curve (without self-intersections), and let G be the doubly-connected domain that is bounded by E1 and E2, seeFig. 4. We call such a G a ring domain. For ring domains one can give an alternative definition

G

E1

E2

E2

G

E1

Figure 4: Ring domains

of condenser capacity. Let

u(z) :=

∫

log(z − t) dµ∗(t) + c1.

E. B. Saff 194

The complex function u is locally analytic but not single-valued inG (notice that there is no modulussign in the integral). Moreover, if we fix t and let z move along a simple closed counterclockwiseoriented curve in G that encircles E1, say, then the imaginary part of log(z − t) increases by 2π,for t ∈ E1, while for t ∈ E2 it returns to the original value. Since µ∗1 and µ∗2 are unit measures, itfollows that the function φ : z → w = exp(u(z)) is analytic and single-valued. Moreover, it can beshown to be one-to-one in G. By its definition, φ satisfies

log |φ| = −Uµ∗+ c1 = 0 on E1; log |φ| = −Uµ∗

+ c1 = c1 + c2 on E2.

Therefore φ maps G conformally onto the annulus 1 < |w| < ec1+c2 .It is known from the theory of conformal mapping that, for a ring domain G, there exists

unique R > 1, called the modulus of G (we denote it by mod(G)), such that G can be mappedconformally onto the annulus 1 < |w| < R. We have thus shown that

cap(E1, E2) = 1/ log(mod(G)). (5.4)

We remark that if G1 ⊃ G2 are two ring domains, then mod(G1) ≥ mod(G2).

Example 5.2. Let E1, E2 be as above, and assume that E2 is the R-th level curve for E1.That is, |Φ(z)| = R for z ∈ E2, where Φ maps conformally the unbounded component of C \ E1

onto |w| > 1. In particular, Φ maps the corresponding ring domain G onto the annulus 1 < |w| < R,and we conclude that mod(G) = R (so that cap(E1, E2) = 1/ logR). Applying this to the configu-ration of Example 5.1, we see that Φ(z) = z/r1, so that R = r2/r1, and we obtain again (5.3).

We now turn to rational approximation. Let E ⊂ C be compact. We denote by Rn the collectionof all rational functions of the form R = P/Q, where P , Q are polynomials of degree at most n,and Q has no zeros in E. For f ∈ A(E), let

rn(f ;E) = rn(f) := infr∈Rn

‖f − r‖E

be the error in best approximation of f by rational functions from Rn. Clearly, since polynomialsare rational functions, we have (cf. (4.1)) rn(f) ≤ en(f). A basic theorem regarding the rate ofrational approximation was proved by Walsh [W, Ch.IX]. Following is a special case of this theorem.

Theorem 5.3. (Walsh) Let E be a single Jordan arc or curve and let f be analytic on asimply connected domain D ⊃ E. Then

lim supn→∞

rn(f)1/n ≤ exp(−1/cap(E, ∂D)). (5.5)

The proof of (5.5) follows the same ideas as the proof of inequality (4.3). Let Γ be a contourin D \ E that is arbitrarily close to ∂D. Let µ∗ = µ∗1 − µ∗2 be the equilibrium measure for the

condenser (E,Γ). For any n, let α(n)1 , . . . , α

(n)n be equally spaced on E (with respect to µ∗1) and let

β(n)1 , . . . , β

(n)n be equally spaced on Γ (with respect to µ∗2). Then one can show that the rational


functions rn(z) with zeros at the α(n)i ’s and poles at the β

(n)i ’s satisfy

maxE

|rn|min

Γ|rn|

1/n

→ e−1/cap(E,Γ). (5.6)

Let Rn = pn−1/qn be the rational function with poles at the β(n)i ’s that interpolates f at the points

α(n)i ’s. Then the Hermite formula (cf. (4.2)) takes the following form:

f(z) −Rn(z) =1

2πi

∫

Γ

rn(z)

rn(t)

f(t)

t− zdt, z inside Γ,

and it follows from (5.6) that

lim supn→∞

rn(f)1/n ≤ lim supn→∞

‖f −Rn‖1/nE ≤ e−1/cap(E,Γ).

Letting Γ approach ∂D, we get the result.

Remarks.(a) Unlike in the polynomial approximation, no rate of convergence of rn(f) to 0 can ensure thata function f ∈ C(E) is analytic somewhere beyond E.(b) One can construct a function for which equality holds in (5.5), so that this bound is sharp.Such a function necessarily has a singularity at every point of ∂D; otherwise f would be analyticin a larger domain, so that the corresponding condenser capacity will become smaller. In view ofTheorem 5.3, this would violate the assumed equality in (5.5).

Although sharp, the bound (5.5) is unsatisfactory, in the following sense. Assume, for example,that E is connected and has a connected complement, and let ΓR, R > 1, be a level curve for E.Let f be a function that is analytic in the domain D bounded by ΓR and such that the equalityholds in (5.5). According to Example 5.2, we then obtain that

lim supn→∞

rn(f)1/n =1

R. (5.7)

By Remark (b) above, such f must have singularities on ΓR. Hence (recall Remark following The-orem 4.1) the relation (5.7) holds with rn(f) replaced by en(f). But the family Rn contains Pn

and it is much more rich than Pn — it depends on 2n + 1 parameters while Pn depends only onn+1 parameters. One would expect, therefore, that at least for a subsequence of n’s, rn(f) behavesasymptotically like e2n(f). This was a motivation for the following conjecture.

Conjecture. (A.A. Gonchar) Let E be a compact set and f be analytic in an open setD containingE. Then

lim infn→∞

rn(f ;E)1/n ≤ exp(−2/cap(E, ∂D)). (5.8)

This conjecture was proved by O. Parfenov [Pa] for the case when E is a continuum with connectedcomplement and in the general case by V. Prokhorov [P]; they used a very different method —the so-called “AAK Theory” (cf. [Y]). However this method is not constructive, and it remains a

E. B. Saff 196

challenging problem to find such a method. Yet, potential theory can be used to obtain boundslike (5.8) in the stronger form

limn→∞

rn(f ;E)1/n = exp(−2/cap(E, ∂D))

for some important subclasses of analytic functions, such as Markov functions (cf. [Gon]) andfunctions with a finite number of algebraic branch-points (cf. [St]).

6 Logarithmic Potentials with External Fields

Let E be a closed (not necessarily compact) subset of C and let w(z) be a nonnegative weight onE. We define a new “distance function” on E, replacing |z − t| by |z − t|w(z)w(t). This gives riseto weighted versions of logarithmic capacity, transfinite diameter and Chebyshev constant.

Weighted capacity: cap(w,E).As before, let M(E) denote the collection of all unit measures supported on E. We set

Q := log1

w

and call it the external field. Consider the modified energy integral for µ ∈ M(E):

Iw(µ) :=

∫ ∫

log1

|z − t|w(z)w(t)dµ(z) dµ(t) (6.1)

=

∫ ∫

log1

|z − t| dµ(z) dµ(t) + 2

∫

Q(z) dµ(z)

and letVw := inf

µ∈M(E)Iw(µ).

The weighted capacity is defined by

cap(w,E) := e−Vw .

In the sequel, we assume that w satisfies the following conditions:

(i) w > 0 on a subset of positive logarithmic capacity;

(ii) w is continuous (or, more generally, upper semi-continuous);

(iii) if E is unbounded, then |z|w(z) → 0 as |z| → ∞, z ∈ E.

Under these restrictions on w, there exists a unique measure µw ∈ M(E), called the weightedequilibrium measure, such that

I(µw) = Vw.

The above integral (6.1) can be interpreted as the total energy of the unit charge µ, in the presenceof the external field Q (in this electrostatics interpretation, the field is actually 2Q). Since this fieldhas a strong repelling effect near points where w = 0 (i.e., Q = ∞), assumption (iii) physicallymeans that, for the equilibrium distribution, no charge occurs near ∞. In other words, the support


supp(µw) of µw is necessarily compact. However, unlike the unweighted case, the support need notlie entirely on ∂∞E and, in fact, it can be quite an arbitrary closed subset of E. Determining thisset is one of the most important aspects of weighted potential theory.

Weighted transfinite diameter: τ(w,E).Let

δn(w) := maxz1,...,zn∈E

∏

1≤i<j≤n

|zi − zj |w(zi)w(zj)

2/n(n−1)

.

Points z(n)1 , . . . , z

(n)n at which the maximum is attained are called weighted Fekete points. The

corresponding Fekete polynomial is the monic polynomial with all its zeros at these points.As in the unweighted case, the sequence δn(w) is decreasing, so one can define

τ(w,E) := limn→∞

δn(w),

which we call the weighted transfinite diameter of E.

Weighted Chebyshev constant: cheb(w,E).Let

tn(w) := minp∈Pn−1

‖wn(z)(zn − p(z))‖E .

Then the weighted Chebyshev constant is defined by

cheb(w,E) := limn→∞

tn(w)1/n.

The following theorem generalizes the fundamental results stated in Theorem 1.18.

Theorem 6.1 (Generalized Fundamental Theorem). Let E be a closed set of positivecapacity. Assume that w satisfies the conditions (i)–(iii) and let Q = log(1/w). Then

cap(w,E) = τ(w,E) = cheb(w,E) exp

(

−∫

Qdµw

)

.

Moreover, weighted Fekete points have asymptotic distribution µw as n→ ∞, and weighted Feketepolynomials are asymptotically optimal for the weighted Chebyshev problem.

How can one find µw?In most applications, the weight w is continuous and the set E is regular. Recall that the

latter means that the classical (unweighted) equilibrium potential for E is equal to VE everywhereon E, not just quasi-everywhere. Under these assumptions, the equilibrium measure µ = µw ischaracterized by the conditions that µ ∈ M(E), I(µ) <∞ and, for some constant cw, the followingvariational conditions hold:

Uµ +Q = cw on S(µ) = supp(µ)Uµ +Q ≥ cw on E.

(6.2)

On integrating (against µ = µw) the first condition, we obtain that the constant is given by

cw = I(µw) +

∫

Qdµw = Vw −∫

Qdµw.

E. B. Saff 198

When trying to find µw, an essential step (and a nontrivial problem in its own right!) is to determinethe support S(µw) := supp(µw). There are several methods by which S(µw) can be numericallyapproximated, but they are complicated from the computational point of view. Therefore, knowingproperties of the support can be useful and we list some of them.

Properties of the support S(µw)(a) The sup norm of weighted polynomials “lives” on S(µw). That is, for any n and for anypolynomial Pn of degree at most n, there holds

‖wnPn‖E = ‖wnPn‖S(µw).

(b) Let K be a compact subset of E of positive capacity, and define

F (K) := log cap(K) −∫

KQdµK ,

where µK is the classical (unweighted) equilibrium measure for K. This so-called F-functional ofMhaskar and Saff is often a helpful tool in finding S(µw). Since cap(K) and µK remain the sameif we replace K by ∂∞K, we obtain that F (K) = F (∂∞K). It turns out that the outer boundaryof S(µw) maximizes the F-functional:

maxK

F (K) = F (∂∞S(µK)).

This result is especially useful when E is a real interval and Q is convex. It is then easy to derivefrom (6.2) that S(µw) is an interval. Thus, to find the support, one merely needs to maximize F (K)only over intervals K ⊂ E, which amounts to a standard calculus problem for the determination ofthe endpoints of S(µw).

(c) S(µw) is the set of weighted polynomial peaking points; that is, if w is continuous andE is of positive capacity at each of its points, then z belongs to S(µw) iff for every disk Dr(z)there is a weighted polynomial wnPn that attains its maximum modulus only in Dr(z) (cf. [ST,Sec. IV.1]).

Example 6.2. Incomplete polynomialsFor the study of incomplete polynomials of type θ on the interval E = [0, 1]; that is, poly-

nomials of the form p(x) =∑n

k=s akxk where s/n ≥ θ, the appropriate external field is Q(x) =

log(1/w(x)) = − θ1−θ log x which is convex. Maximizing the F-functional one gets S(µw) = [θ2, 1].

(For details, see [ST, Sec. IV.1].)

Example 6.3. Freud WeightsHere E = R and w(x) = exp(−|x|α). Hence Q(x) = |x|α is convex provided that α > 1, and

we obtain Sw = [−aα, aα], where aα can be given explicitly in terms of the Gamma function. (Ac-tually, this result also holds for all α > 0; see [ST, Sec. IV.1].) For example, when α = 2, we getSw = [−1, 1].

The Generalized Weierstrass Approximation Problem mentioned in problem (v) of the intro-duction states the following: For E ⊂ R closed, w : E → [0,∞), characterize those functions f


continuous on E that are uniform limits on E of some sequence of weighted polynomials (wnPn),degPn ≤ n.

To attack this problem, we begin with a crucial observation. Let E be a closed subset of R

whose complement is regular and w(x) be continuous on E. Then we have the following weightedanalogue of the Bernstein-Walsh lemma:

|wn(x)Pn(x)| ≤ ‖wnPn‖S(µw) exp(−n(Uµw(x) +Q(x) − cw)), x ∈ E \ S(µw).

With the aid of (6.2) and a variant of the Stone-Weierstrass theorem (cf. [ST]), one can show thatif a sequence (wn(x)Pn(x)), degPn ≤ n, converges uniformly on E, then it tends to 0 for everyx ∈ E \ S(µw).

Thus, if some f ∈ C(E) is a uniform limit on E of such a sequence as n → ∞, it must vanishon E \ S(µw). The converse is not true, in general, but it is true in many important cases, suchas for incomplete polynomials where the weight w(x) = xθ/(1−θ) on [0, 1] and for Freud weightsw(x) = exp(−|x|α), α > 1, on R. The latter fact provided an essential ingredient in resolvingproblem (iv) of the introduction (see [LuSa] and [LMS]).

For the case when E is a real interval and Q = log(1/w) is convex on E, this author conjecturedand Totik [To] has proved that, more generally, any f ∈ C(E) that vanishes on E \ S(µw) is theuniform limit on E of some sequence of weighted polynomials (wnPn), degPn ≤ n.

References

[B] T. Bagby, The modulus of a plane condenser, J. Math. Mech., 17 (1976) 315–329.

[F] G. Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy.Irish Acad. Sect. A(1), 76 (1976) 1–6.

[Ga] D. Gaier, Lectures on Complex Approximation, Birkhauser, Boston Inc., Boston, MA, 1987.

[Gon] A.A. Gonchar, The rate of rational approximation of analytic functions. (Russian) Modernproblems of mathematics. Differential equations, mathematical analysis and their applica-tions. Trudy Mat. Inst. Steklov, 166 (1984), 52–60.

[La] N.S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Heidelberg, 1972.

[Lo] G.G. Lorentz, Approximation by incomplete polynomials (problems and results). In E.B.Saff and R.S. Varga, editors, Pade and Rational Approximations: Theory and Applications,289–302, Academic Press, New York, 1977.

[LuSa] D.S. Lubinsky and E.B. Saff, Uniform and mean approximation by certain weighted polyno-mials, with applications, Constr. Approx., 4 (1988) 21–64.

[LMS] D.S. Lubinsky, H.N. Mhaskar, and E.B. Saff, Freud’s conjecture for exponential weights,Bull. Amer. Math. Soc., 15 (1986) 217–221.

[Pa] O.G. Parfenov, Estimates of singular numbers of the Carleson embedding operator, Mat. Sb.(N.S.), 131(173) (1986) 501–518; English transl. in Math. USSR-Sb., 59 (1988) 497–514.

[P] V.A. Prokhorov, Rational approximation of analytic functions, Mat. Sb, 184 (1993) 3–32.English transl. Russian Acad. Sci. Sb. Math. 78 (1994) 139–164.

E. B. Saff 200

[R1] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge,1995.

[R2] T. Ransford, Computation of logarithmic capacity, Comp. Methods Function Theory, (toappear).

[ST] E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, NewYork, 1997.

[St] H. Stahl, General convergence results for rational approximation, in: Approximation TheoryVI, volume 2, C.K. Chui, L.L. Schumaker and J.D. Ward (eds.), Academic Press, Boston,(1989) 605–634.

[Sz] G. Szego, Orthogonal Polynomials, volume 23 of Colloquium Publications, Amer. Math. Soc.,Providence, R.I., 1975.

[To] V. Totik, Weighted polynomial approximation for convex external fields, Constr. Approx., 16(2000) 261–281.

[Ts] M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

[W] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Plane,volume 20 of Colloquium Publications, Amer. Math. Soc., Providence, R.I., 1960.

[Y] N. Young, An Introduction to Hilbert Space, Cambridge University Press, Cambridge, 1988.

Edward B. SaffCenter for Constructive ApproximationDepartment of MathematicsVanderbilt UniversityNashville TN 37240 [email protected]

http://www.math.vanderbilt.edu/~esaff